Course description: The course is about the non-classical calculus of probability which is behind Quantum Physics. ( Read this short summary written in a "Q&A" form about the essence of Quantum Physics.) The emphasis will be on the mathematical, information-theoretical and philosophical aspects (but not directly on physics). In the first part of the course the necessary mathematicals tools are introduced, while in the second part, we describe quantum physical systems from an informational theoretical point of view, we discuss concepts like entanglement, we treat some "paradoxes" (such as the "EPR" paradox), go over some information technological protocols (such as dense coding) and finish with a short introduction to quantum computers.

Quantum information theory is the main motivations behind our study. The aim of the course is "merely" to give a sound foundation to the understanding of quantum phenomena; so the mentioned information technological applications - like quantum computers - though treated to some extent, will not form the core of the course.

Topics:

• The non-distributive structure of events in the quantum word. Elements of a non-classical probability theory I.
• Elements of a non-classical probability theory II: the convex structure of probability functions, intrinsic uncertainties versus lack of knowledge.
• Linear algebra brush up: complementary subspaces and projections, linear maps and their matrices in different bases, trace, determinant, eigenvalues and eigenspaces.
• Advanced linear algebra: spectral decomposition and spectral calculus (only in finite dimensions), complex vector spaces with scalar product, operator norm, the adjoint of an operator.
• The word of linear operators: normal, self-adjoint, unitary, and positive operators and their spectral characterization. Spectral calculus of normal operators.
• The convex cone of positive operators. Operator inequalities and trace inequalities. The body of density operators.
• Quantum probability theory: Hilbert lattices and Gleason's theorem (without proof).
• The general framework of quantum physics. States and measurements.
• Finite level quantum systems, quantum bit and their information capacity.
• Composite (bipartite) systems. The role of tensorial products.
• Entanglement. Quantum telepathy games and the Bell-inequalities.
• Physical operations as completely positive maps. The no-cloning theorem. Dense-coding.
• Quantum gates and quantum computers. Example of an algorithm for a quantum computer (either Grover's search algorithm or Shor's algorithm for factorizing integers).